Rosa's FloatVsDoubleBenchmark

Percentage Accurate: 69.7% → 99.6%

Time: 7.7s

Alternatives: 24

Speedup: 8.5×

• 47.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1)))
   (+
    x1
    (+
     (+
      (+
       (+
        (*
         (+
          (* (* (* 2.0 x1) t_2) (- t_2 3.0))
          (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
         t_1)
        (* t_0 t_2))
       (* (* x1 x1) x1))
      x1)
     (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	return x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x1, x2)
use fmin_fmax_functions
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = (3.0d0 * x1) * x1
    t_1 = (x1 * x1) + 1.0d0
    t_2 = ((t_0 + (2.0d0 * x2)) - x1) / t_1
    code = x1 + (((((((((2.0d0 * x1) * t_2) * (t_2 - 3.0d0)) + ((x1 * x1) * ((4.0d0 * t_2) - 6.0d0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0d0 * (((t_0 - (2.0d0 * x2)) - x1) / t_1)))
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	return x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
}

Python

def code(x1, x2):
	t_0 = (3.0 * x1) * x1
	t_1 = (x1 * x1) + 1.0
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1
	return x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	return Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
end

MATLAB

function tmp = code(x1, x2)
	t_0 = (3.0 * x1) * x1;
	t_1 = (x1 * x1) + 1.0;
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	tmp = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1)))
   (+
    x1
    (+
     (+
      (+
       (+
        (*
         (+
          (* (* (* 2.0 x1) t_2) (- t_2 3.0))
          (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
         t_1)
        (* t_0 t_2))
       (* (* x1 x1) x1))
      x1)
     (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	return x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x1, x2)
use fmin_fmax_functions
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = (3.0d0 * x1) * x1
    t_1 = (x1 * x1) + 1.0d0
    t_2 = ((t_0 + (2.0d0 * x2)) - x1) / t_1
    code = x1 + (((((((((2.0d0 * x1) * t_2) * (t_2 - 3.0d0)) + ((x1 * x1) * ((4.0d0 * t_2) - 6.0d0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0d0 * (((t_0 - (2.0d0 * x2)) - x1) / t_1)))
end function

Java

public static double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	return x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
}

Python

def code(x1, x2):
	t_0 = (3.0 * x1) * x1
	t_1 = (x1 * x1) + 1.0
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1
	return x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	return Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
end

MATLAB

function tmp = code(x1, x2)
	t_0 = (3.0 * x1) * x1;
	t_1 = (x1 * x1) + 1.0;
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	tmp = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_2) (- t_2 3.0))
                (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
               t_1)
              (* t_0 t_2))
             (* (* x1 x1) x1))
            x1)
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))))
   (if (<= t_3 INFINITY)
     t_3
     (* (* (* x1 x1) (* x1 x1)) (- 6.0 (* -8.0 (/ x2 (* x1 x1))))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_3 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))));
	}
	return tmp;
}

Java

public static double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))));
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (3.0 * x1) * x1
	t_1 = (x1 * x1) + 1.0
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1
	t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)))
	tmp = 0
	if t_3 <= math.inf:
		tmp = t_3
	else:
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))))
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	tmp = 0.0
	if (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(x1 * x1) * Float64(x1 * x1)) * Float64(6.0 - Float64(-8.0 * Float64(x2 / Float64(x1 * x1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (3.0 * x1) * x1;
	t_1 = (x1 * x1) + 1.0;
	t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	tmp = 0.0;
	if (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, Infinity], t$95$3, N[(N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(6.0 - N[(-8.0 * N[(x2 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.4%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   if +inf.0 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 0.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x2 around inf

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \color{blue}{\frac{x2}{{x1}^{2}}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{\color{blue}{{x1}^{2}}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{{x1}^{\color{blue}{2}}}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right) \]

      4. lift-*.f64 100.0

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right) \]

   7. Applied rewrites 100.0%

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \color{blue}{\frac{x2}{x1 \cdot x1}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := \left(3 \cdot x1\right) \cdot x1\\ t_2 := x1 \cdot x1 + 1\\ t_3 := 3 \cdot \left(x1 \cdot x1\right) - x1\\ t_4 := 1 + x1 \cdot x1\\ t_5 := \frac{\left(t\_1 + 2 \cdot x2\right) - x1}{t\_2}\\ t_6 := \frac{t\_3}{t\_4}\\ \mathbf{if}\;x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_5\right) \cdot \left(t\_5 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_5 - 6\right)\right) \cdot t\_2 + t\_1 \cdot t\_5\right) + t\_0\right) + x1\right) + 3 \cdot \frac{\left(t\_1 - 2 \cdot x2\right) - x1}{t\_2}\right) \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot t\_3}{t\_4}, \mathsf{fma}\left(3, t\_6, \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, \mathsf{fma}\left(t\_4, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{t\_4} - \left(3 + \frac{x1}{t\_4}\right)\right) \cdot t\_3\right)}{t\_4}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_6 - 6\right)\right), t\_0\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* x1 x1) x1))
        (t_1 (* (* 3.0 x1) x1))
        (t_2 (+ (* x1 x1) 1.0))
        (t_3 (- (* 3.0 (* x1 x1)) x1))
        (t_4 (+ 1.0 (* x1 x1)))
        (t_5 (/ (- (+ t_1 (* 2.0 x2)) x1) t_2))
        (t_6 (/ t_3 t_4)))
   (if (<=
        (+
         x1
         (+
          (+
           (+
            (+
             (*
              (+
               (* (* (* 2.0 x1) t_5) (- t_5 3.0))
               (* (* x1 x1) (- (* 4.0 t_5) 6.0)))
              t_2)
             (* t_1 t_5))
            t_0)
           x1)
          (* 3.0 (/ (- (- t_1 (* 2.0 x2)) x1) t_2))))
        INFINITY)
     (fma
      2.0
      x1
      (fma
       3.0
       (/ (* (* x1 x1) t_3) t_4)
       (fma
        3.0
        t_6
        (fma
         x2
         (- (* x1 (- (* 8.0 x2) 12.0)) 6.0)
         (fma
          t_4
          (fma
           2.0
           (/
            (* x1 (* (- (* 3.0 (/ (* x1 x1) t_4)) (+ 3.0 (/ x1 t_4))) t_3))
            t_4)
           (* (* x1 x1) (- (* 4.0 t_6) 6.0)))
          t_0)))))
     (* (* (* x1 x1) (* x1 x1)) (- 6.0 (* -8.0 (/ x2 (* x1 x1))))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (3.0 * x1) * x1;
	double t_2 = (x1 * x1) + 1.0;
	double t_3 = (3.0 * (x1 * x1)) - x1;
	double t_4 = 1.0 + (x1 * x1);
	double t_5 = ((t_1 + (2.0 * x2)) - x1) / t_2;
	double t_6 = t_3 / t_4;
	double tmp;
	if ((x1 + (((((((((2.0 * x1) * t_5) * (t_5 - 3.0)) + ((x1 * x1) * ((4.0 * t_5) - 6.0))) * t_2) + (t_1 * t_5)) + t_0) + x1) + (3.0 * (((t_1 - (2.0 * x2)) - x1) / t_2)))) <= ((double) INFINITY)) {
		tmp = fma(2.0, x1, fma(3.0, (((x1 * x1) * t_3) / t_4), fma(3.0, t_6, fma(x2, ((x1 * ((8.0 * x2) - 12.0)) - 6.0), fma(t_4, fma(2.0, ((x1 * (((3.0 * ((x1 * x1) / t_4)) - (3.0 + (x1 / t_4))) * t_3)) / t_4), ((x1 * x1) * ((4.0 * t_6) - 6.0))), t_0)))));
	} else {
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * x1)
	t_1 = Float64(Float64(3.0 * x1) * x1)
	t_2 = Float64(Float64(x1 * x1) + 1.0)
	t_3 = Float64(Float64(3.0 * Float64(x1 * x1)) - x1)
	t_4 = Float64(1.0 + Float64(x1 * x1))
	t_5 = Float64(Float64(Float64(t_1 + Float64(2.0 * x2)) - x1) / t_2)
	t_6 = Float64(t_3 / t_4)
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_5) * Float64(t_5 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_5) - 6.0))) * t_2) + Float64(t_1 * t_5)) + t_0) + x1) + Float64(3.0 * Float64(Float64(Float64(t_1 - Float64(2.0 * x2)) - x1) / t_2)))) <= Inf)
		tmp = fma(2.0, x1, fma(3.0, Float64(Float64(Float64(x1 * x1) * t_3) / t_4), fma(3.0, t_6, fma(x2, Float64(Float64(x1 * Float64(Float64(8.0 * x2) - 12.0)) - 6.0), fma(t_4, fma(2.0, Float64(Float64(x1 * Float64(Float64(Float64(3.0 * Float64(Float64(x1 * x1) / t_4)) - Float64(3.0 + Float64(x1 / t_4))) * t_3)) / t_4), Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_6) - 6.0))), t_0)))));
	else
		tmp = Float64(Float64(Float64(x1 * x1) * Float64(x1 * x1)) * Float64(6.0 - Float64(-8.0 * Float64(x2 / Float64(x1 * x1)))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(3.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t$95$1 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 / t$95$4), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(t$95$5 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$5), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(t$95$1 * t$95$5), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$1 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(2.0 * x1 + N[(3.0 * N[(N[(N[(x1 * x1), $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$4), $MachinePrecision] + N[(3.0 * t$95$6 + N[(x2 * N[(N[(x1 * N[(N[(8.0 * x2), $MachinePrecision] - 12.0), $MachinePrecision]), $MachinePrecision] - 6.0), $MachinePrecision] + N[(t$95$4 * N[(2.0 * N[(N[(x1 * N[(N[(N[(3.0 * N[(N[(x1 * x1), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] - N[(3.0 + N[(x1 / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$6), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(6.0 - N[(-8.0 * N[(x2 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.4%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right) \]

      4. lower-*.f64 95.3

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right) \]

   7. Applied rewrites 95.3%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, x1 \cdot \left(8 \cdot x2 - 12\right) - 6, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right) \]

   if +inf.0 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 0.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x2 around inf

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \color{blue}{\frac{x2}{{x1}^{2}}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{\color{blue}{{x1}^{2}}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{{x1}^{\color{blue}{2}}}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right) \]

      4. lift-*.f64 100.0

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right) \]

   7. Applied rewrites 100.0%

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \color{blue}{\frac{x2}{x1 \cdot x1}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \left(3 \cdot x1\right) \cdot x1\\ t_3 := \frac{\left(t\_2 + 2 \cdot x2\right) - x1}{t\_1}\\ t_4 := t\_2 \cdot t\_3\\ t_5 := \left(\left(2 \cdot x1\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right)\\ t_6 := 3 \cdot \frac{\left(t\_2 - 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(\left(\left(\left(t\_5 + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_3 - 6\right)\right) \cdot t\_1 + t\_4\right) + t\_0\right) + x1\right) + t\_6\right) \leq \infty:\\ \;\;\;\;x1 + \left(\left(\left(\left(\left(t\_5 + \left(x1 \cdot x1\right) \cdot 6\right) \cdot t\_1 + t\_4\right) + t\_0\right) + x1\right) + t\_6\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* x1 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (* (* 3.0 x1) x1))
        (t_3 (/ (- (+ t_2 (* 2.0 x2)) x1) t_1))
        (t_4 (* t_2 t_3))
        (t_5 (* (* (* 2.0 x1) t_3) (- t_3 3.0)))
        (t_6 (* 3.0 (/ (- (- t_2 (* 2.0 x2)) x1) t_1))))
   (if (<=
        (+
         x1
         (+
          (+
           (+ (+ (* (+ t_5 (* (* x1 x1) (- (* 4.0 t_3) 6.0))) t_1) t_4) t_0)
           x1)
          t_6))
        INFINITY)
     (+ x1 (+ (+ (+ (+ (* (+ t_5 (* (* x1 x1) 6.0)) t_1) t_4) t_0) x1) t_6))
     (* (* (* x1 x1) (* x1 x1)) (- 6.0 (* -8.0 (/ x2 (* x1 x1))))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = (3.0 * x1) * x1;
	double t_3 = ((t_2 + (2.0 * x2)) - x1) / t_1;
	double t_4 = t_2 * t_3;
	double t_5 = ((2.0 * x1) * t_3) * (t_3 - 3.0);
	double t_6 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	double tmp;
	if ((x1 + ((((((t_5 + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + t_4) + t_0) + x1) + t_6)) <= ((double) INFINITY)) {
		tmp = x1 + ((((((t_5 + ((x1 * x1) * 6.0)) * t_1) + t_4) + t_0) + x1) + t_6);
	} else {
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))));
	}
	return tmp;
}

Java

public static double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = (3.0 * x1) * x1;
	double t_3 = ((t_2 + (2.0 * x2)) - x1) / t_1;
	double t_4 = t_2 * t_3;
	double t_5 = ((2.0 * x1) * t_3) * (t_3 - 3.0);
	double t_6 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	double tmp;
	if ((x1 + ((((((t_5 + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + t_4) + t_0) + x1) + t_6)) <= Double.POSITIVE_INFINITY) {
		tmp = x1 + ((((((t_5 + ((x1 * x1) * 6.0)) * t_1) + t_4) + t_0) + x1) + t_6);
	} else {
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))));
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (x1 * x1) * x1
	t_1 = (x1 * x1) + 1.0
	t_2 = (3.0 * x1) * x1
	t_3 = ((t_2 + (2.0 * x2)) - x1) / t_1
	t_4 = t_2 * t_3
	t_5 = ((2.0 * x1) * t_3) * (t_3 - 3.0)
	t_6 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1)
	tmp = 0
	if (x1 + ((((((t_5 + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + t_4) + t_0) + x1) + t_6)) <= math.inf:
		tmp = x1 + ((((((t_5 + ((x1 * x1) * 6.0)) * t_1) + t_4) + t_0) + x1) + t_6)
	else:
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))))
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(3.0 * x1) * x1)
	t_3 = Float64(Float64(Float64(t_2 + Float64(2.0 * x2)) - x1) / t_1)
	t_4 = Float64(t_2 * t_3)
	t_5 = Float64(Float64(Float64(2.0 * x1) * t_3) * Float64(t_3 - 3.0))
	t_6 = Float64(3.0 * Float64(Float64(Float64(t_2 - Float64(2.0 * x2)) - x1) / t_1))
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(t_5 + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_3) - 6.0))) * t_1) + t_4) + t_0) + x1) + t_6)) <= Inf)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(t_5 + Float64(Float64(x1 * x1) * 6.0)) * t_1) + t_4) + t_0) + x1) + t_6));
	else
		tmp = Float64(Float64(Float64(x1 * x1) * Float64(x1 * x1)) * Float64(6.0 - Float64(-8.0 * Float64(x2 / Float64(x1 * x1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * x1) * x1;
	t_1 = (x1 * x1) + 1.0;
	t_2 = (3.0 * x1) * x1;
	t_3 = ((t_2 + (2.0 * x2)) - x1) / t_1;
	t_4 = t_2 * t_3;
	t_5 = ((2.0 * x1) * t_3) * (t_3 - 3.0);
	t_6 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	tmp = 0.0;
	if ((x1 + ((((((t_5 + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + t_4) + t_0) + x1) + t_6)) <= Inf)
		tmp = x1 + ((((((t_5 + ((x1 * x1) * 6.0)) * t_1) + t_4) + t_0) + x1) + t_6);
	else
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(3.0 * N[(N[(N[(t$95$2 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(N[(N[(N[(N[(t$95$5 + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$3), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision], Infinity], N[(x1 + N[(N[(N[(N[(N[(N[(t$95$5 + N[(N[(x1 * x1), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(6.0 - N[(-8.0 * N[(x2 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.4%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around inf

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{6}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 95.8%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{6}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   if +inf.0 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 0.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x2 around inf

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \color{blue}{\frac{x2}{{x1}^{2}}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{\color{blue}{{x1}^{2}}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{{x1}^{\color{blue}{2}}}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right) \]

      4. lift-*.f64 100.0

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right) \]

   7. Applied rewrites 100.0%

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \color{blue}{\frac{x2}{x1 \cdot x1}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \left(3 \cdot x1\right) \cdot x1\\ t_3 := \frac{\left(t\_2 + 2 \cdot x2\right) - x1}{t\_1}\\ t_4 := \left(\left(2 \cdot x1\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right)\\ t_5 := 3 \cdot \frac{\left(t\_2 - 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(\left(\left(\left(t\_4 + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_3 - 6\right)\right) \cdot t\_1 + t\_2 \cdot t\_3\right) + t\_0\right) + x1\right) + t\_5\right) \leq \infty:\\ \;\;\;\;x1 + \left(\left(\left(\left(\left(t\_4 + \left(x1 \cdot x1\right) \cdot 6\right) \cdot t\_1 + 9 \cdot \left(x1 \cdot x1\right)\right) + t\_0\right) + x1\right) + t\_5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* x1 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (* (* 3.0 x1) x1))
        (t_3 (/ (- (+ t_2 (* 2.0 x2)) x1) t_1))
        (t_4 (* (* (* 2.0 x1) t_3) (- t_3 3.0)))
        (t_5 (* 3.0 (/ (- (- t_2 (* 2.0 x2)) x1) t_1))))
   (if (<=
        (+
         x1
         (+
          (+
           (+
            (+ (* (+ t_4 (* (* x1 x1) (- (* 4.0 t_3) 6.0))) t_1) (* t_2 t_3))
            t_0)
           x1)
          t_5))
        INFINITY)
     (+
      x1
      (+
       (+ (+ (+ (* (+ t_4 (* (* x1 x1) 6.0)) t_1) (* 9.0 (* x1 x1))) t_0) x1)
       t_5))
     (* (* (* x1 x1) (* x1 x1)) (- 6.0 (* -8.0 (/ x2 (* x1 x1))))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = (3.0 * x1) * x1;
	double t_3 = ((t_2 + (2.0 * x2)) - x1) / t_1;
	double t_4 = ((2.0 * x1) * t_3) * (t_3 - 3.0);
	double t_5 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	double tmp;
	if ((x1 + ((((((t_4 + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + (t_2 * t_3)) + t_0) + x1) + t_5)) <= ((double) INFINITY)) {
		tmp = x1 + ((((((t_4 + ((x1 * x1) * 6.0)) * t_1) + (9.0 * (x1 * x1))) + t_0) + x1) + t_5);
	} else {
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))));
	}
	return tmp;
}

Java

public static double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = (3.0 * x1) * x1;
	double t_3 = ((t_2 + (2.0 * x2)) - x1) / t_1;
	double t_4 = ((2.0 * x1) * t_3) * (t_3 - 3.0);
	double t_5 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	double tmp;
	if ((x1 + ((((((t_4 + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + (t_2 * t_3)) + t_0) + x1) + t_5)) <= Double.POSITIVE_INFINITY) {
		tmp = x1 + ((((((t_4 + ((x1 * x1) * 6.0)) * t_1) + (9.0 * (x1 * x1))) + t_0) + x1) + t_5);
	} else {
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))));
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (x1 * x1) * x1
	t_1 = (x1 * x1) + 1.0
	t_2 = (3.0 * x1) * x1
	t_3 = ((t_2 + (2.0 * x2)) - x1) / t_1
	t_4 = ((2.0 * x1) * t_3) * (t_3 - 3.0)
	t_5 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1)
	tmp = 0
	if (x1 + ((((((t_4 + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + (t_2 * t_3)) + t_0) + x1) + t_5)) <= math.inf:
		tmp = x1 + ((((((t_4 + ((x1 * x1) * 6.0)) * t_1) + (9.0 * (x1 * x1))) + t_0) + x1) + t_5)
	else:
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))))
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(3.0 * x1) * x1)
	t_3 = Float64(Float64(Float64(t_2 + Float64(2.0 * x2)) - x1) / t_1)
	t_4 = Float64(Float64(Float64(2.0 * x1) * t_3) * Float64(t_3 - 3.0))
	t_5 = Float64(3.0 * Float64(Float64(Float64(t_2 - Float64(2.0 * x2)) - x1) / t_1))
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(t_4 + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_3) - 6.0))) * t_1) + Float64(t_2 * t_3)) + t_0) + x1) + t_5)) <= Inf)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(t_4 + Float64(Float64(x1 * x1) * 6.0)) * t_1) + Float64(9.0 * Float64(x1 * x1))) + t_0) + x1) + t_5));
	else
		tmp = Float64(Float64(Float64(x1 * x1) * Float64(x1 * x1)) * Float64(6.0 - Float64(-8.0 * Float64(x2 / Float64(x1 * x1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * x1) * x1;
	t_1 = (x1 * x1) + 1.0;
	t_2 = (3.0 * x1) * x1;
	t_3 = ((t_2 + (2.0 * x2)) - x1) / t_1;
	t_4 = ((2.0 * x1) * t_3) * (t_3 - 3.0);
	t_5 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	tmp = 0.0;
	if ((x1 + ((((((t_4 + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + (t_2 * t_3)) + t_0) + x1) + t_5)) <= Inf)
		tmp = x1 + ((((((t_4 + ((x1 * x1) * 6.0)) * t_1) + (9.0 * (x1 * x1))) + t_0) + x1) + t_5);
	else
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(3.0 * N[(N[(N[(t$95$2 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(N[(N[(N[(N[(t$95$4 + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$3), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision], Infinity], N[(x1 + N[(N[(N[(N[(N[(N[(t$95$4 + N[(N[(x1 * x1), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(6.0 - N[(-8.0 * N[(x2 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.4%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around inf

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{6}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 95.8%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{6}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   5. Taylor expanded in x1 around inf

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + \color{blue}{9 \cdot {x1}^{2}}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + 9 \cdot \color{blue}{{x1}^{2}}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. pow2 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + 9 \cdot \left(x1 \cdot \color{blue}{x1}\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      3. lift-*.f64 95.8

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + 9 \cdot \left(x1 \cdot \color{blue}{x1}\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   7. Applied rewrites 95.8%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + \color{blue}{9 \cdot \left(x1 \cdot x1\right)}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   if +inf.0 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 0.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x2 around inf

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \color{blue}{\frac{x2}{{x1}^{2}}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{\color{blue}{{x1}^{2}}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{{x1}^{\color{blue}{2}}}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right) \]

      4. lift-*.f64 100.0

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right) \]

   7. Applied rewrites 100.0%

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \color{blue}{\frac{x2}{x1 \cdot x1}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \left(3 \cdot x1\right) \cdot x1\\ t_3 := \frac{\left(t\_2 + 2 \cdot x2\right) - x1}{t\_1}\\ t_4 := 3 \cdot \frac{\left(t\_2 - 2 \cdot x2\right) - x1}{t\_1}\\ \mathbf{if}\;x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_3\right) \cdot \left(t\_3 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_3 - 6\right)\right) \cdot t\_1 + t\_2 \cdot t\_3\right) + t\_0\right) + x1\right) + t\_4\right) \leq \infty:\\ \;\;\;\;x1 + \left(\left(\left(\left(\left(\left(4 \cdot \frac{x1 \cdot x2}{1 + x1 \cdot x1}\right) \cdot \left(\frac{2 \cdot x2}{t\_1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot t\_1 + 9 \cdot \left(x1 \cdot x1\right)\right) + t\_0\right) + x1\right) + t\_4\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* x1 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (* (* 3.0 x1) x1))
        (t_3 (/ (- (+ t_2 (* 2.0 x2)) x1) t_1))
        (t_4 (* 3.0 (/ (- (- t_2 (* 2.0 x2)) x1) t_1))))
   (if (<=
        (+
         x1
         (+
          (+
           (+
            (+
             (*
              (+
               (* (* (* 2.0 x1) t_3) (- t_3 3.0))
               (* (* x1 x1) (- (* 4.0 t_3) 6.0)))
              t_1)
             (* t_2 t_3))
            t_0)
           x1)
          t_4))
        INFINITY)
     (+
      x1
      (+
       (+
        (+
         (+
          (*
           (+
            (*
             (* 4.0 (/ (* x1 x2) (+ 1.0 (* x1 x1))))
             (- (/ (* 2.0 x2) t_1) 3.0))
            (* (* x1 x1) 6.0))
           t_1)
          (* 9.0 (* x1 x1)))
         t_0)
        x1)
       t_4))
     (* (* (* x1 x1) (* x1 x1)) (- 6.0 (* -8.0 (/ x2 (* x1 x1))))))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = (3.0 * x1) * x1;
	double t_3 = ((t_2 + (2.0 * x2)) - x1) / t_1;
	double t_4 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	double tmp;
	if ((x1 + (((((((((2.0 * x1) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + (t_2 * t_3)) + t_0) + x1) + t_4)) <= ((double) INFINITY)) {
		tmp = x1 + ((((((((4.0 * ((x1 * x2) / (1.0 + (x1 * x1)))) * (((2.0 * x2) / t_1) - 3.0)) + ((x1 * x1) * 6.0)) * t_1) + (9.0 * (x1 * x1))) + t_0) + x1) + t_4);
	} else {
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))));
	}
	return tmp;
}

Java

public static double code(double x1, double x2) {
	double t_0 = (x1 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = (3.0 * x1) * x1;
	double t_3 = ((t_2 + (2.0 * x2)) - x1) / t_1;
	double t_4 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	double tmp;
	if ((x1 + (((((((((2.0 * x1) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + (t_2 * t_3)) + t_0) + x1) + t_4)) <= Double.POSITIVE_INFINITY) {
		tmp = x1 + ((((((((4.0 * ((x1 * x2) / (1.0 + (x1 * x1)))) * (((2.0 * x2) / t_1) - 3.0)) + ((x1 * x1) * 6.0)) * t_1) + (9.0 * (x1 * x1))) + t_0) + x1) + t_4);
	} else {
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))));
	}
	return tmp;
}

Python

def code(x1, x2):
	t_0 = (x1 * x1) * x1
	t_1 = (x1 * x1) + 1.0
	t_2 = (3.0 * x1) * x1
	t_3 = ((t_2 + (2.0 * x2)) - x1) / t_1
	t_4 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1)
	tmp = 0
	if (x1 + (((((((((2.0 * x1) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + (t_2 * t_3)) + t_0) + x1) + t_4)) <= math.inf:
		tmp = x1 + ((((((((4.0 * ((x1 * x2) / (1.0 + (x1 * x1)))) * (((2.0 * x2) / t_1) - 3.0)) + ((x1 * x1) * 6.0)) * t_1) + (9.0 * (x1 * x1))) + t_0) + x1) + t_4)
	else:
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))))
	return tmp

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(3.0 * x1) * x1)
	t_3 = Float64(Float64(Float64(t_2 + Float64(2.0 * x2)) - x1) / t_1)
	t_4 = Float64(3.0 * Float64(Float64(Float64(t_2 - Float64(2.0 * x2)) - x1) / t_1))
	tmp = 0.0
	if (Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_3) * Float64(t_3 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_3) - 6.0))) * t_1) + Float64(t_2 * t_3)) + t_0) + x1) + t_4)) <= Inf)
		tmp = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(4.0 * Float64(Float64(x1 * x2) / Float64(1.0 + Float64(x1 * x1)))) * Float64(Float64(Float64(2.0 * x2) / t_1) - 3.0)) + Float64(Float64(x1 * x1) * 6.0)) * t_1) + Float64(9.0 * Float64(x1 * x1))) + t_0) + x1) + t_4));
	else
		tmp = Float64(Float64(Float64(x1 * x1) * Float64(x1 * x1)) * Float64(6.0 - Float64(-8.0 * Float64(x2 / Float64(x1 * x1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x1, x2)
	t_0 = (x1 * x1) * x1;
	t_1 = (x1 * x1) + 1.0;
	t_2 = (3.0 * x1) * x1;
	t_3 = ((t_2 + (2.0 * x2)) - x1) / t_1;
	t_4 = 3.0 * (((t_2 - (2.0 * x2)) - x1) / t_1);
	tmp = 0.0;
	if ((x1 + (((((((((2.0 * x1) * t_3) * (t_3 - 3.0)) + ((x1 * x1) * ((4.0 * t_3) - 6.0))) * t_1) + (t_2 * t_3)) + t_0) + x1) + t_4)) <= Inf)
		tmp = x1 + ((((((((4.0 * ((x1 * x2) / (1.0 + (x1 * x1)))) * (((2.0 * x2) / t_1) - 3.0)) + ((x1 * x1) * 6.0)) * t_1) + (9.0 * (x1 * x1))) + t_0) + x1) + t_4);
	else
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(3.0 * N[(N[(N[(t$95$2 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$3), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision], Infinity], N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(4.0 * N[(N[(x1 * x2), $MachinePrecision] / N[(1.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 * x2), $MachinePrecision] / t$95$1), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(9.0 * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] + x1), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(6.0 - N[(-8.0 * N[(x2 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < +inf.0

   1. Initial program 99.4%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around inf

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{6}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 95.8%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \color{blue}{6}\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   5. Taylor expanded in x2 around inf

      \[\leadsto x1 + \left(\left(\left(\left(\left(\color{blue}{\left(4 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}}\right)} \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   6. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(4 \cdot \frac{x1 \cdot x2}{1 + x1 \cdot \color{blue}{x1}}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(4 \cdot \color{blue}{\frac{x1 \cdot x2}{1 + x1 \cdot x1}}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(4 \cdot \frac{x1 \cdot x2}{1 + x1 \cdot \color{blue}{x1}}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      4. lift-+.f64 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(4 \cdot \frac{x1 \cdot x2}{1 + \color{blue}{x1 \cdot x1}}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(4 \cdot \frac{x1 \cdot x2}{\color{blue}{1 + x1 \cdot x1}}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      6. lift-*.f64 95.9

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(4 \cdot \frac{x1 \cdot x2}{\color{blue}{1} + x1 \cdot x1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   7. Applied rewrites 95.9%

      \[\leadsto x1 + \left(\left(\left(\left(\left(\color{blue}{\left(4 \cdot \frac{x1 \cdot x2}{1 + x1 \cdot x1}\right)} \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   8. Taylor expanded in x1 around inf

      \[\leadsto x1 + \left(\left(\left(\left(\left(\left(4 \cdot \frac{x1 \cdot x2}{1 + x1 \cdot x1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + \color{blue}{9 \cdot {x1}^{2}}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(4 \cdot \frac{x1 \cdot x2}{1 + x1 \cdot x1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + 9 \cdot \color{blue}{{x1}^{2}}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      2. pow2 N/A

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(4 \cdot \frac{x1 \cdot x2}{1 + x1 \cdot x1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + 9 \cdot \left(x1 \cdot \color{blue}{x1}\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

      3. lift-*.f64 95.8

         \[\leadsto x1 + \left(\left(\left(\left(\left(\left(4 \cdot \frac{x1 \cdot x2}{1 + x1 \cdot x1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + 9 \cdot \left(x1 \cdot \color{blue}{x1}\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   10. Applied rewrites 95.8%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(4 \cdot \frac{x1 \cdot x2}{1 + x1 \cdot x1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + \color{blue}{9 \cdot \left(x1 \cdot x1\right)}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   11. Taylor expanded in x1 around 0

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(4 \cdot \frac{x1 \cdot x2}{1 + x1 \cdot x1}\right) \cdot \left(\frac{\color{blue}{2 \cdot x2}}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + 9 \cdot \left(x1 \cdot x1\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 95.8

          \[\leadsto x1 + \left(\left(\left(\left(\left(\left(4 \cdot \frac{x1 \cdot x2}{1 + x1 \cdot x1}\right) \cdot \left(\frac{2 \cdot \color{blue}{x2}}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + 9 \cdot \left(x1 \cdot x1\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   13. Applied rewrites 95.8%

       \[\leadsto x1 + \left(\left(\left(\left(\left(\left(4 \cdot \frac{x1 \cdot x2}{1 + x1 \cdot x1}\right) \cdot \left(\frac{\color{blue}{2 \cdot x2}}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot 6\right) \cdot \left(x1 \cdot x1 + 1\right) + 9 \cdot \left(x1 \cdot x1\right)\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   if +inf.0 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 0.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x2 around inf

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \color{blue}{\frac{x2}{{x1}^{2}}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{\color{blue}{{x1}^{2}}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{{x1}^{\color{blue}{2}}}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right) \]

      4. lift-*.f64 100.0

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right) \]

   7. Applied rewrites 100.0%

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \color{blue}{\frac{x2}{x1 \cdot x1}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 95.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot x2 - 3\\ \mathbf{if}\;x1 \leq -6600:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot t\_0\right)\\ \mathbf{elif}\;x1 \leq 19:\\ \;\;\;\;\mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 + \mathsf{fma}\left(-1, \frac{1 - 2 \cdot \left(1 - -3 \cdot t\_0\right)}{x1}, 4 \cdot t\_0\right)}{x1}}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (- (* 2.0 x2) 3.0)))
   (if (<= x1 -6600.0)
     (* (* x1 x1) (- (+ 9.0 (* x1 (- (* 6.0 x1) 3.0))) (* -4.0 t_0)))
     (if (<= x1 19.0)
       (fma
        2.0
        x1
        (fma -6.0 x2 (fma -3.0 x1 (* x2 (fma -12.0 x1 (* 8.0 (* x1 x2)))))))
       (*
        (* (* x1 x1) (* x1 x1))
        (-
         6.0
         (*
          1.0
          (/
           (-
            3.0
            (*
             1.0
             (/
              (+
               9.0
               (fma
                -1.0
                (/ (- 1.0 (* 2.0 (- 1.0 (* -3.0 t_0)))) x1)
                (* 4.0 t_0)))
              x1)))
           x1))))))))

C

double code(double x1, double x2) {
	double t_0 = (2.0 * x2) - 3.0;
	double tmp;
	if (x1 <= -6600.0) {
		tmp = (x1 * x1) * ((9.0 + (x1 * ((6.0 * x1) - 3.0))) - (-4.0 * t_0));
	} else if (x1 <= 19.0) {
		tmp = fma(2.0, x1, fma(-6.0, x2, fma(-3.0, x1, (x2 * fma(-12.0, x1, (8.0 * (x1 * x2)))))));
	} else {
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (1.0 * ((3.0 - (1.0 * ((9.0 + fma(-1.0, ((1.0 - (2.0 * (1.0 - (-3.0 * t_0)))) / x1), (4.0 * t_0))) / x1))) / x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(2.0 * x2) - 3.0)
	tmp = 0.0
	if (x1 <= -6600.0)
		tmp = Float64(Float64(x1 * x1) * Float64(Float64(9.0 + Float64(x1 * Float64(Float64(6.0 * x1) - 3.0))) - Float64(-4.0 * t_0)));
	elseif (x1 <= 19.0)
		tmp = fma(2.0, x1, fma(-6.0, x2, fma(-3.0, x1, Float64(x2 * fma(-12.0, x1, Float64(8.0 * Float64(x1 * x2)))))));
	else
		tmp = Float64(Float64(Float64(x1 * x1) * Float64(x1 * x1)) * Float64(6.0 - Float64(1.0 * Float64(Float64(3.0 - Float64(1.0 * Float64(Float64(9.0 + fma(-1.0, Float64(Float64(1.0 - Float64(2.0 * Float64(1.0 - Float64(-3.0 * t_0)))) / x1), Float64(4.0 * t_0))) / x1))) / x1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]}, If[LessEqual[x1, -6600.0], N[(N[(x1 * x1), $MachinePrecision] * N[(N[(9.0 + N[(x1 * N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-4.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 19.0], N[(2.0 * x1 + N[(-6.0 * x2 + N[(-3.0 * x1 + N[(x2 * N[(-12.0 * x1 + N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(6.0 - N[(1.0 * N[(N[(3.0 - N[(1.0 * N[(N[(9.0 + N[(-1.0 * N[(N[(1.0 - N[(2.0 * N[(1.0 - N[(-3.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision] + N[(4.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -6600

   1. Initial program 32.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto {x1}^{2} \cdot \color{blue}{\left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{2} \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - \color{blue}{-4 \cdot \left(2 \cdot x2 - 3\right)}\right) \]

      2. pow2 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - \color{blue}{-4} \cdot \left(2 \cdot x2 - 3\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - \color{blue}{-4} \cdot \left(2 \cdot x2 - 3\right)\right) \]

      4. lower--.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \color{blue}{\left(2 \cdot x2 - 3\right)}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(\color{blue}{2 \cdot x2} - 3\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot \color{blue}{x2} - 3\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      11. lift-*.f64 93.5

          \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - \color{blue}{3}\right)\right) \]

   7. Applied rewrites 93.5%

      \[\leadsto \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right)} \]

   if -6600 < x1 < 19

   1. Initial program 99.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 86.0%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, -3 \cdot x1 + x2 \cdot \left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

      5. lift-*.f64 97.5

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

   10. Applied rewrites 97.5%

       \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

   if 19 < x1

   1. Initial program 47.1%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + \left(-1 \cdot \frac{1 + -2 \cdot \left(1 + 3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1} + 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 + \mathsf{fma}\left(-1, \frac{1 - 2 \cdot \left(1 - -3 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}, 4 \cdot \left(2 \cdot x2 - 3\right)\right)}{x1}}{x1}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 95.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x1 \leq -6600:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right)\\ \mathbf{elif}\;x1 \leq 19:\\ \;\;\;\;\mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{\mathsf{fma}\left(-1, \frac{17 + -12 \cdot x2}{x1}, 8 \cdot x2\right) - 3}{x1}}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x1 -6600.0)
   (*
    (* x1 x1)
    (- (+ 9.0 (* x1 (- (* 6.0 x1) 3.0))) (* -4.0 (- (* 2.0 x2) 3.0))))
   (if (<= x1 19.0)
     (fma
      2.0
      x1
      (fma -6.0 x2 (fma -3.0 x1 (* x2 (fma -12.0 x1 (* 8.0 (* x1 x2)))))))
     (*
      (pow x1 4.0)
      (+
       6.0
       (*
        -1.0
        (/
         (+
          3.0
          (*
           -1.0
           (/ (- (fma -1.0 (/ (+ 17.0 (* -12.0 x2)) x1) (* 8.0 x2)) 3.0) x1)))
         x1)))))))

C

double code(double x1, double x2) {
	double tmp;
	if (x1 <= -6600.0) {
		tmp = (x1 * x1) * ((9.0 + (x1 * ((6.0 * x1) - 3.0))) - (-4.0 * ((2.0 * x2) - 3.0)));
	} else if (x1 <= 19.0) {
		tmp = fma(2.0, x1, fma(-6.0, x2, fma(-3.0, x1, (x2 * fma(-12.0, x1, (8.0 * (x1 * x2)))))));
	} else {
		tmp = pow(x1, 4.0) * (6.0 + (-1.0 * ((3.0 + (-1.0 * ((fma(-1.0, ((17.0 + (-12.0 * x2)) / x1), (8.0 * x2)) - 3.0) / x1))) / x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x1 <= -6600.0)
		tmp = Float64(Float64(x1 * x1) * Float64(Float64(9.0 + Float64(x1 * Float64(Float64(6.0 * x1) - 3.0))) - Float64(-4.0 * Float64(Float64(2.0 * x2) - 3.0))));
	elseif (x1 <= 19.0)
		tmp = fma(2.0, x1, fma(-6.0, x2, fma(-3.0, x1, Float64(x2 * fma(-12.0, x1, Float64(8.0 * Float64(x1 * x2)))))));
	else
		tmp = Float64((x1 ^ 4.0) * Float64(6.0 + Float64(-1.0 * Float64(Float64(3.0 + Float64(-1.0 * Float64(Float64(fma(-1.0, Float64(Float64(17.0 + Float64(-12.0 * x2)) / x1), Float64(8.0 * x2)) - 3.0) / x1))) / x1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x1, -6600.0], N[(N[(x1 * x1), $MachinePrecision] * N[(N[(9.0 + N[(x1 * N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-4.0 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 19.0], N[(2.0 * x1 + N[(-6.0 * x2 + N[(-3.0 * x1 + N[(x2 * N[(-12.0 * x1 + N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x1, 4.0], $MachinePrecision] * N[(6.0 + N[(-1.0 * N[(N[(3.0 + N[(-1.0 * N[(N[(N[(-1.0 * N[(N[(17.0 + N[(-12.0 * x2), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision] + N[(8.0 * x2), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -6600

   1. Initial program 32.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto {x1}^{2} \cdot \color{blue}{\left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{2} \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - \color{blue}{-4 \cdot \left(2 \cdot x2 - 3\right)}\right) \]

      2. pow2 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - \color{blue}{-4} \cdot \left(2 \cdot x2 - 3\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - \color{blue}{-4} \cdot \left(2 \cdot x2 - 3\right)\right) \]

      4. lower--.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \color{blue}{\left(2 \cdot x2 - 3\right)}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(\color{blue}{2 \cdot x2} - 3\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot \color{blue}{x2} - 3\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      11. lift-*.f64 93.5

          \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - \color{blue}{3}\right)\right) \]

   7. Applied rewrites 93.5%

      \[\leadsto \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right)} \]

   if -6600 < x1 < 19

   1. Initial program 99.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 86.0%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, -3 \cdot x1 + x2 \cdot \left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

      5. lift-*.f64 97.5

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

   10. Applied rewrites 97.5%

       \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

   if 19 < x1

   1. Initial program 47.1%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around -inf

      \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{\left(-1 \cdot \frac{17 + -12 \cdot x2}{x1} + 8 \cdot x2\right) - 3}{x1}}{x1}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 94.4%

      \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{\mathsf{fma}\left(-1, \frac{17 + -12 \cdot x2}{x1}, 8 \cdot x2\right) - 3}{x1}}{x1}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 95.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \left(2 \cdot x2 - 3\right)\\ \mathbf{if}\;x1 \leq -6600:\\ \;\;\;\;\left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - t\_0\right)\\ \mathbf{elif}\;x1 \leq 19:\\ \;\;\;\;\mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - t\_0}{x1}}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* -4.0 (- (* 2.0 x2) 3.0))))
   (if (<= x1 -6600.0)
     (* (* x1 x1) (- (+ 9.0 (* x1 (- (* 6.0 x1) 3.0))) t_0))
     (if (<= x1 19.0)
       (fma
        2.0
        x1
        (fma -6.0 x2 (fma -3.0 x1 (* x2 (fma -12.0 x1 (* 8.0 (* x1 x2)))))))
       (*
        (* (* x1 x1) (* x1 x1))
        (- 6.0 (* 1.0 (/ (- 3.0 (* 1.0 (/ (- 9.0 t_0) x1))) x1))))))))

C

double code(double x1, double x2) {
	double t_0 = -4.0 * ((2.0 * x2) - 3.0);
	double tmp;
	if (x1 <= -6600.0) {
		tmp = (x1 * x1) * ((9.0 + (x1 * ((6.0 * x1) - 3.0))) - t_0);
	} else if (x1 <= 19.0) {
		tmp = fma(2.0, x1, fma(-6.0, x2, fma(-3.0, x1, (x2 * fma(-12.0, x1, (8.0 * (x1 * x2)))))));
	} else {
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (1.0 * ((3.0 - (1.0 * ((9.0 - t_0) / x1))) / x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(-4.0 * Float64(Float64(2.0 * x2) - 3.0))
	tmp = 0.0
	if (x1 <= -6600.0)
		tmp = Float64(Float64(x1 * x1) * Float64(Float64(9.0 + Float64(x1 * Float64(Float64(6.0 * x1) - 3.0))) - t_0));
	elseif (x1 <= 19.0)
		tmp = fma(2.0, x1, fma(-6.0, x2, fma(-3.0, x1, Float64(x2 * fma(-12.0, x1, Float64(8.0 * Float64(x1 * x2)))))));
	else
		tmp = Float64(Float64(Float64(x1 * x1) * Float64(x1 * x1)) * Float64(6.0 - Float64(1.0 * Float64(Float64(3.0 - Float64(1.0 * Float64(Float64(9.0 - t_0) / x1))) / x1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(-4.0 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -6600.0], N[(N[(x1 * x1), $MachinePrecision] * N[(N[(9.0 + N[(x1 * N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 19.0], N[(2.0 * x1 + N[(-6.0 * x2 + N[(-3.0 * x1 + N[(x2 * N[(-12.0 * x1 + N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(6.0 - N[(1.0 * N[(N[(3.0 - N[(1.0 * N[(N[(9.0 - t$95$0), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -6600

   1. Initial program 32.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto {x1}^{2} \cdot \color{blue}{\left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{2} \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - \color{blue}{-4 \cdot \left(2 \cdot x2 - 3\right)}\right) \]

      2. pow2 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - \color{blue}{-4} \cdot \left(2 \cdot x2 - 3\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - \color{blue}{-4} \cdot \left(2 \cdot x2 - 3\right)\right) \]

      4. lower--.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \color{blue}{\left(2 \cdot x2 - 3\right)}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(\color{blue}{2 \cdot x2} - 3\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot \color{blue}{x2} - 3\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      11. lift-*.f64 93.5

          \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - \color{blue}{3}\right)\right) \]

   7. Applied rewrites 93.5%

      \[\leadsto \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right)} \]

   if -6600 < x1 < 19

   1. Initial program 99.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 86.0%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, -3 \cdot x1 + x2 \cdot \left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

      5. lift-*.f64 97.5

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

   10. Applied rewrites 97.5%

       \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

   if 19 < x1

   1. Initial program 47.1%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 95.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right)\\ \mathbf{if}\;x1 \leq -6600:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 19:\\ \;\;\;\;\mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          (* x1 x1)
          (- (+ 9.0 (* x1 (- (* 6.0 x1) 3.0))) (* -4.0 (- (* 2.0 x2) 3.0))))))
   (if (<= x1 -6600.0)
     t_0
     (if (<= x1 19.0)
       (fma
        2.0
        x1
        (fma -6.0 x2 (fma -3.0 x1 (* x2 (fma -12.0 x1 (* 8.0 (* x1 x2)))))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * ((9.0 + (x1 * ((6.0 * x1) - 3.0))) - (-4.0 * ((2.0 * x2) - 3.0)));
	double tmp;
	if (x1 <= -6600.0) {
		tmp = t_0;
	} else if (x1 <= 19.0) {
		tmp = fma(2.0, x1, fma(-6.0, x2, fma(-3.0, x1, (x2 * fma(-12.0, x1, (8.0 * (x1 * x2)))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * Float64(Float64(9.0 + Float64(x1 * Float64(Float64(6.0 * x1) - 3.0))) - Float64(-4.0 * Float64(Float64(2.0 * x2) - 3.0))))
	tmp = 0.0
	if (x1 <= -6600.0)
		tmp = t_0;
	elseif (x1 <= 19.0)
		tmp = fma(2.0, x1, fma(-6.0, x2, fma(-3.0, x1, Float64(x2 * fma(-12.0, x1, Float64(8.0 * Float64(x1 * x2)))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * N[(N[(9.0 + N[(x1 * N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-4.0 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -6600.0], t$95$0, If[LessEqual[x1, 19.0], N[(2.0 * x1 + N[(-6.0 * x2 + N[(-3.0 * x1 + N[(x2 * N[(-12.0 * x1 + N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -6600 or 19 < x1

   1. Initial program 40.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto {x1}^{2} \cdot \color{blue}{\left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{2} \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - \color{blue}{-4 \cdot \left(2 \cdot x2 - 3\right)}\right) \]

      2. pow2 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - \color{blue}{-4} \cdot \left(2 \cdot x2 - 3\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - \color{blue}{-4} \cdot \left(2 \cdot x2 - 3\right)\right) \]

      4. lower--.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \color{blue}{\left(2 \cdot x2 - 3\right)}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(\color{blue}{2 \cdot x2} - 3\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot \color{blue}{x2} - 3\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      11. lift-*.f64 93.9

          \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - \color{blue}{3}\right)\right) \]

   7. Applied rewrites 93.9%

      \[\leadsto \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right)} \]

   if -6600 < x1 < 19

   1. Initial program 99.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 86.0%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, -3 \cdot x1 + x2 \cdot \left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

      5. lift-*.f64 97.5

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]

   10. Applied rewrites 97.5%

       \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, \mathsf{fma}\left(-3, x1, x2 \cdot \mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 95.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right)\\ \mathbf{if}\;x1 \leq -6600:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 19:\\ \;\;\;\;\mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0
         (*
          (* x1 x1)
          (- (+ 9.0 (* x1 (- (* 6.0 x1) 3.0))) (* -4.0 (- (* 2.0 x2) 3.0))))))
   (if (<= x1 -6600.0)
     t_0
     (if (<= x1 19.0)
       (fma
        2.0
        x1
        (fma -3.0 x1 (* x2 (- (fma -12.0 x1 (* 8.0 (* x1 x2))) 6.0))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = (x1 * x1) * ((9.0 + (x1 * ((6.0 * x1) - 3.0))) - (-4.0 * ((2.0 * x2) - 3.0)));
	double tmp;
	if (x1 <= -6600.0) {
		tmp = t_0;
	} else if (x1 <= 19.0) {
		tmp = fma(2.0, x1, fma(-3.0, x1, (x2 * (fma(-12.0, x1, (8.0 * (x1 * x2))) - 6.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(x1 * x1) * Float64(Float64(9.0 + Float64(x1 * Float64(Float64(6.0 * x1) - 3.0))) - Float64(-4.0 * Float64(Float64(2.0 * x2) - 3.0))))
	tmp = 0.0
	if (x1 <= -6600.0)
		tmp = t_0;
	elseif (x1 <= 19.0)
		tmp = fma(2.0, x1, fma(-3.0, x1, Float64(x2 * Float64(fma(-12.0, x1, Float64(8.0 * Float64(x1 * x2))) - 6.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(x1 * x1), $MachinePrecision] * N[(N[(9.0 + N[(x1 * N[(N[(6.0 * x1), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-4.0 * N[(N[(2.0 * x2), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -6600.0], t$95$0, If[LessEqual[x1, 19.0], N[(2.0 * x1 + N[(-3.0 * x1 + N[(x2 * N[(N[(-12.0 * x1 + N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -6600 or 19 < x1

   1. Initial program 40.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto {x1}^{2} \cdot \color{blue}{\left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{2} \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - \color{blue}{-4 \cdot \left(2 \cdot x2 - 3\right)}\right) \]

      2. pow2 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - \color{blue}{-4} \cdot \left(2 \cdot x2 - 3\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - \color{blue}{-4} \cdot \left(2 \cdot x2 - 3\right)\right) \]

      4. lower--.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \color{blue}{\left(2 \cdot x2 - 3\right)}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(\color{blue}{2 \cdot x2} - 3\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot \color{blue}{x2} - 3\right)\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right) \]

      11. lift-*.f64 93.9

          \[\leadsto \left(x1 \cdot x1\right) \cdot \left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - \color{blue}{3}\right)\right) \]

   7. Applied rewrites 93.9%

      \[\leadsto \left(x1 \cdot x1\right) \cdot \color{blue}{\left(\left(9 + x1 \cdot \left(6 \cdot x1 - 3\right)\right) - -4 \cdot \left(2 \cdot x2 - 3\right)\right)} \]

   if -6600 < x1 < 19

   1. Initial program 99.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 86.0%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -3 \cdot x1 + x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right) \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)\right) \]

      6. lift-*.f64 97.3

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)\right) \]

   10. Applied rewrites 97.3%

       \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 94.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right)\\ \mathbf{if}\;x1 \leq -11000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1.1 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* (* x1 x1) (* x1 x1)) (- 6.0 (* -8.0 (/ x2 (* x1 x1)))))))
   (if (<= x1 -11000.0)
     t_0
     (if (<= x1 1.1e+29)
       (fma
        2.0
        x1
        (fma -3.0 x1 (* x2 (- (fma -12.0 x1 (* 8.0 (* x1 x2))) 6.0))))
       t_0))))

C

double code(double x1, double x2) {
	double t_0 = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))));
	double tmp;
	if (x1 <= -11000.0) {
		tmp = t_0;
	} else if (x1 <= 1.1e+29) {
		tmp = fma(2.0, x1, fma(-3.0, x1, (x2 * (fma(-12.0, x1, (8.0 * (x1 * x2))) - 6.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(Float64(x1 * x1) * Float64(x1 * x1)) * Float64(6.0 - Float64(-8.0 * Float64(x2 / Float64(x1 * x1)))))
	tmp = 0.0
	if (x1 <= -11000.0)
		tmp = t_0;
	elseif (x1 <= 1.1e+29)
		tmp = fma(2.0, x1, fma(-3.0, x1, Float64(x2 * Float64(fma(-12.0, x1, Float64(8.0 * Float64(x1 * x2))) - 6.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(6.0 - N[(-8.0 * N[(x2 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -11000.0], t$95$0, If[LessEqual[x1, 1.1e+29], N[(2.0 * x1 + N[(-3.0 * x1 + N[(x2 * N[(N[(-12.0 * x1 + N[(8.0 * N[(x1 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -11000 or 1.1000000000000001e29 < x1

   1. Initial program 37.6%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x2 around inf

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \color{blue}{\frac{x2}{{x1}^{2}}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{\color{blue}{{x1}^{2}}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{{x1}^{\color{blue}{2}}}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right) \]

      4. lift-*.f64 94.9

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right) \]

   7. Applied rewrites 94.9%

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \color{blue}{\frac{x2}{x1 \cdot x1}}\right) \]

   if -11000 < x1 < 1.1000000000000001e29

   1. Initial program 99.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 83.8%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -3 \cdot x1 + x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right) \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(\left(-12 \cdot x1 + 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)\right) \]

      6. lift-*.f64 94.7

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)\right) \]

   10. Applied rewrites 94.7%

       \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(\mathsf{fma}\left(-12, x1, 8 \cdot \left(x1 \cdot x2\right)\right) - 6\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 90.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 1\right)\right)\\ t_1 := \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right)\\ \mathbf{if}\;x1 \leq -11000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq -2.9 \cdot 10^{-254}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1.6 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot -12 - 3\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 1.1 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma -6.0 x2 (* x1 (- (* x2 (- (* 8.0 x2) 12.0)) 1.0))))
        (t_1 (* (* (* x1 x1) (* x1 x1)) (- 6.0 (* -8.0 (/ x2 (* x1 x1)))))))
   (if (<= x1 -11000.0)
     t_1
     (if (<= x1 -2.9e-254)
       t_0
       (if (<= x1 1.6e-173)
         (fma 2.0 x1 (fma -6.0 x2 (* x1 (- (* x2 -12.0) 3.0))))
         (if (<= x1 1.1e+29) t_0 t_1))))))

C

double code(double x1, double x2) {
	double t_0 = fma(-6.0, x2, (x1 * ((x2 * ((8.0 * x2) - 12.0)) - 1.0)));
	double t_1 = ((x1 * x1) * (x1 * x1)) * (6.0 - (-8.0 * (x2 / (x1 * x1))));
	double tmp;
	if (x1 <= -11000.0) {
		tmp = t_1;
	} else if (x1 <= -2.9e-254) {
		tmp = t_0;
	} else if (x1 <= 1.6e-173) {
		tmp = fma(2.0, x1, fma(-6.0, x2, (x1 * ((x2 * -12.0) - 3.0))));
	} else if (x1 <= 1.1e+29) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(-6.0, x2, Float64(x1 * Float64(Float64(x2 * Float64(Float64(8.0 * x2) - 12.0)) - 1.0)))
	t_1 = Float64(Float64(Float64(x1 * x1) * Float64(x1 * x1)) * Float64(6.0 - Float64(-8.0 * Float64(x2 / Float64(x1 * x1)))))
	tmp = 0.0
	if (x1 <= -11000.0)
		tmp = t_1;
	elseif (x1 <= -2.9e-254)
		tmp = t_0;
	elseif (x1 <= 1.6e-173)
		tmp = fma(2.0, x1, fma(-6.0, x2, Float64(x1 * Float64(Float64(x2 * -12.0) - 3.0))));
	elseif (x1 <= 1.1e+29)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(-6.0 * x2 + N[(x1 * N[(N[(x2 * N[(N[(8.0 * x2), $MachinePrecision] - 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(6.0 - N[(-8.0 * N[(x2 / N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -11000.0], t$95$1, If[LessEqual[x1, -2.9e-254], t$95$0, If[LessEqual[x1, 1.6e-173], N[(2.0 * x1 + N[(-6.0 * x2 + N[(x1 * N[(N[(x2 * -12.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 1.1e+29], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -11000 or 1.1000000000000001e29 < x1

   1. Initial program 37.6%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x2 around inf

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \color{blue}{\frac{x2}{{x1}^{2}}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{\color{blue}{{x1}^{2}}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{{x1}^{\color{blue}{2}}}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right) \]

      4. lift-*.f64 94.9

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \frac{x2}{x1 \cdot x1}\right) \]

   7. Applied rewrites 94.9%

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - -8 \cdot \color{blue}{\frac{x2}{x1 \cdot x1}}\right) \]

   if -11000 < x1 < -2.9e-254 or 1.6e-173 < x1 < 1.1000000000000001e29

   1. Initial program 99.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto -6 \cdot x2 + \color{blue}{x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.3%

      \[\leadsto \mathsf{fma}\left(-6, \color{blue}{x2}, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 1\right)\right) \]

   if -2.9e-254 < x1 < 1.6e-173

   1. Initial program 99.5%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 80.9%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot -12 - 3\right)\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 89.9%

       \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot -12 - 3\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 88.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 1\right)\right)\\ t_1 := \left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -8000:\\ \;\;\;\;t\_1 \cdot \left(6 - \frac{3}{x1}\right)\\ \mathbf{elif}\;x1 \leq -2.9 \cdot 10^{-254}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1.6 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot -12 - 3\right)\right)\right)\\ \mathbf{elif}\;x1 \leq 4.1 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;6 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (fma -6.0 x2 (* x1 (- (* x2 (- (* 8.0 x2) 12.0)) 1.0))))
        (t_1 (* (* x1 x1) (* x1 x1))))
   (if (<= x1 -8000.0)
     (* t_1 (- 6.0 (/ 3.0 x1)))
     (if (<= x1 -2.9e-254)
       t_0
       (if (<= x1 1.6e-173)
         (fma 2.0 x1 (fma -6.0 x2 (* x1 (- (* x2 -12.0) 3.0))))
         (if (<= x1 4.1e+30) t_0 (* 6.0 t_1)))))))

C

double code(double x1, double x2) {
	double t_0 = fma(-6.0, x2, (x1 * ((x2 * ((8.0 * x2) - 12.0)) - 1.0)));
	double t_1 = (x1 * x1) * (x1 * x1);
	double tmp;
	if (x1 <= -8000.0) {
		tmp = t_1 * (6.0 - (3.0 / x1));
	} else if (x1 <= -2.9e-254) {
		tmp = t_0;
	} else if (x1 <= 1.6e-173) {
		tmp = fma(2.0, x1, fma(-6.0, x2, (x1 * ((x2 * -12.0) - 3.0))));
	} else if (x1 <= 4.1e+30) {
		tmp = t_0;
	} else {
		tmp = 6.0 * t_1;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = fma(-6.0, x2, Float64(x1 * Float64(Float64(x2 * Float64(Float64(8.0 * x2) - 12.0)) - 1.0)))
	t_1 = Float64(Float64(x1 * x1) * Float64(x1 * x1))
	tmp = 0.0
	if (x1 <= -8000.0)
		tmp = Float64(t_1 * Float64(6.0 - Float64(3.0 / x1)));
	elseif (x1 <= -2.9e-254)
		tmp = t_0;
	elseif (x1 <= 1.6e-173)
		tmp = fma(2.0, x1, fma(-6.0, x2, Float64(x1 * Float64(Float64(x2 * -12.0) - 3.0))));
	elseif (x1 <= 4.1e+30)
		tmp = t_0;
	else
		tmp = Float64(6.0 * t_1);
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(-6.0 * x2 + N[(x1 * N[(N[(x2 * N[(N[(8.0 * x2), $MachinePrecision] - 12.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -8000.0], N[(t$95$1 * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, -2.9e-254], t$95$0, If[LessEqual[x1, 1.6e-173], N[(2.0 * x1 + N[(-6.0 * x2 + N[(x1 * N[(N[(x2 * -12.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x1, 4.1e+30], t$95$0, N[(6.0 * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x1 < -8e3

   1. Initial program 32.6%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x1 around inf

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{\color{blue}{x1}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 88.9

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{x1}\right) \]

   7. Applied rewrites 88.9%

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{\color{blue}{x1}}\right) \]

   if -8e3 < x1 < -2.9e-254 or 1.6e-173 < x1 < 4.10000000000000005e30

   1. Initial program 99.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto -6 \cdot x2 + \color{blue}{x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.1%

      \[\leadsto \mathsf{fma}\left(-6, \color{blue}{x2}, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 1\right)\right) \]

   if -2.9e-254 < x1 < 1.6e-173

   1. Initial program 99.5%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 80.9%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot -12 - 3\right)\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 89.9%

       \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot -12 - 3\right)\right)\right) \]

   if 4.10000000000000005e30 < x1

   1. Initial program 42.6%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around inf

      \[\leadsto \color{blue}{6 \cdot {x1}^{4}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 6 \cdot \color{blue}{{x1}^{4}} \]

      2. sqr-pow N/A

         \[\leadsto 6 \cdot \left({x1}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{x1}^{\left(\frac{4}{2}\right)}}\right) \]

      3. metadata-eval N/A

         \[\leadsto 6 \cdot \left({x1}^{2} \cdot {x1}^{\left(\frac{4}{2}\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto 6 \cdot \left({x1}^{2} \cdot {x1}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 6 \cdot \left({x1}^{2} \cdot \color{blue}{{x1}^{2}}\right) \]

      6. pow2 N/A

         \[\leadsto 6 \cdot \left(\left(x1 \cdot x1\right) \cdot {\color{blue}{x1}}^{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 6 \cdot \left(\left(x1 \cdot x1\right) \cdot {\color{blue}{x1}}^{2}\right) \]

      8. pow2 N/A

         \[\leadsto 6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot \color{blue}{x1}\right)\right) \]

      9. lift-*.f64 91.9

         \[\leadsto 6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot \color{blue}{x1}\right)\right) \]

   4. Applied rewrites 91.9%

      \[\leadsto \color{blue}{6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 80.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -1.5 \cdot 10^{+167}:\\ \;\;\;\;8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + x1 \cdot x1}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot -12 - 3\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_2) (- t_2 3.0))
                (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
               t_1)
              (* t_0 t_2))
             (* (* x1 x1) x1))
            x1)
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))))
   (if (<= t_3 -1.5e+167)
     (* 8.0 (/ (* x1 (* x2 x2)) (+ 1.0 (* x1 x1))))
     (if (<= t_3 2e+47)
       (fma 2.0 x1 (fma -6.0 x2 (* x1 (- (* x2 -12.0) 3.0))))
       (* (* (* x1 x1) (* x1 x1)) (- 6.0 (/ 3.0 x1)))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_3 <= -1.5e+167) {
		tmp = 8.0 * ((x1 * (x2 * x2)) / (1.0 + (x1 * x1)));
	} else if (t_3 <= 2e+47) {
		tmp = fma(2.0, x1, fma(-6.0, x2, (x1 * ((x2 * -12.0) - 3.0))));
	} else {
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (3.0 / x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	tmp = 0.0
	if (t_3 <= -1.5e+167)
		tmp = Float64(8.0 * Float64(Float64(x1 * Float64(x2 * x2)) / Float64(1.0 + Float64(x1 * x1))));
	elseif (t_3 <= 2e+47)
		tmp = fma(2.0, x1, fma(-6.0, x2, Float64(x1 * Float64(Float64(x2 * -12.0) - 3.0))));
	else
		tmp = Float64(Float64(Float64(x1 * x1) * Float64(x1 * x1)) * Float64(6.0 - Float64(3.0 / x1)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.5e+167], N[(8.0 * N[(N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+47], N[(2.0 * x1 + N[(-6.0 * x2 + N[(x1 * N[(N[(x2 * -12.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1.50000000000000006e167

   1. Initial program 99.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around inf

      \[\leadsto \color{blue}{8 \cdot \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 8 \cdot \color{blue}{\frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot {x2}^{2}}{\color{blue}{1 + {x1}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot {x2}^{2}}{\color{blue}{1} + {x1}^{2}} \]

      4. unpow2 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + {x1}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + {x1}^{2}} \]

      6. lower-+.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + \color{blue}{{x1}^{2}}} \]

      7. pow2 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + x1 \cdot \color{blue}{x1}} \]

      8. lift-*.f64 66.1

         \[\leadsto 8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + x1 \cdot \color{blue}{x1}} \]

   4. Applied rewrites 66.1%

      \[\leadsto \color{blue}{8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + x1 \cdot x1}} \]

   if -1.50000000000000006e167 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 2.0000000000000001e47

   1. Initial program 99.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 93.5%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot -12 - 3\right)\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 91.1%

       \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot -12 - 3\right)\right)\right) \]

   if 2.0000000000000001e47 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 48.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x1 around inf

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{\color{blue}{x1}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 76.4

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{x1}\right) \]

   7. Applied rewrites 76.4%

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{\color{blue}{x1}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 80.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -1.5 \cdot 10^{+167}:\\ \;\;\;\;8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + x1 \cdot x1}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(-12 \cdot x1 - 6\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_2) (- t_2 3.0))
                (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
               t_1)
              (* t_0 t_2))
             (* (* x1 x1) x1))
            x1)
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))))
   (if (<= t_3 -1.5e+167)
     (* 8.0 (/ (* x1 (* x2 x2)) (+ 1.0 (* x1 x1))))
     (if (<= t_3 2e+47)
       (fma 2.0 x1 (fma -3.0 x1 (* x2 (- (* -12.0 x1) 6.0))))
       (* (* (* x1 x1) (* x1 x1)) (- 6.0 (/ 3.0 x1)))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_3 <= -1.5e+167) {
		tmp = 8.0 * ((x1 * (x2 * x2)) / (1.0 + (x1 * x1)));
	} else if (t_3 <= 2e+47) {
		tmp = fma(2.0, x1, fma(-3.0, x1, (x2 * ((-12.0 * x1) - 6.0))));
	} else {
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (3.0 / x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	tmp = 0.0
	if (t_3 <= -1.5e+167)
		tmp = Float64(8.0 * Float64(Float64(x1 * Float64(x2 * x2)) / Float64(1.0 + Float64(x1 * x1))));
	elseif (t_3 <= 2e+47)
		tmp = fma(2.0, x1, fma(-3.0, x1, Float64(x2 * Float64(Float64(-12.0 * x1) - 6.0))));
	else
		tmp = Float64(Float64(Float64(x1 * x1) * Float64(x1 * x1)) * Float64(6.0 - Float64(3.0 / x1)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.5e+167], N[(8.0 * N[(N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+47], N[(2.0 * x1 + N[(-3.0 * x1 + N[(x2 * N[(N[(-12.0 * x1), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1.50000000000000006e167

   1. Initial program 99.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around inf

      \[\leadsto \color{blue}{8 \cdot \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 8 \cdot \color{blue}{\frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot {x2}^{2}}{\color{blue}{1 + {x1}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot {x2}^{2}}{\color{blue}{1} + {x1}^{2}} \]

      4. unpow2 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + {x1}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + {x1}^{2}} \]

      6. lower-+.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + \color{blue}{{x1}^{2}}} \]

      7. pow2 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + x1 \cdot \color{blue}{x1}} \]

      8. lift-*.f64 66.1

         \[\leadsto 8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + x1 \cdot \color{blue}{x1}} \]

   4. Applied rewrites 66.1%

      \[\leadsto \color{blue}{8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + x1 \cdot x1}} \]

   if -1.50000000000000006e167 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 2.0000000000000001e47

   1. Initial program 99.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 93.5%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -3 \cdot x1 + x2 \cdot \left(-12 \cdot x1 - 6\right)\right) \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(-12 \cdot x1 - 6\right)\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(-12 \cdot x1 - 6\right)\right)\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(-12 \cdot x1 - 6\right)\right)\right) \]

      4. lower-*.f64 91.1

         \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(-12 \cdot x1 - 6\right)\right)\right) \]

   10. Applied rewrites 91.1%

       \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-3, x1, x2 \cdot \left(-12 \cdot x1 - 6\right)\right)\right) \]

   if 2.0000000000000001e47 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 48.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x1 around inf

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{\color{blue}{x1}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 76.4

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{x1}\right) \]

   7. Applied rewrites 76.4%

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{\color{blue}{x1}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 80.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -1.5 \cdot 10^{+167}:\\ \;\;\;\;8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + x1 \cdot x1}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_2) (- t_2 3.0))
                (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
               t_1)
              (* t_0 t_2))
             (* (* x1 x1) x1))
            x1)
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))))
   (if (<= t_3 -1.5e+167)
     (* 8.0 (/ (* x1 (* x2 x2)) (+ 1.0 (* x1 x1))))
     (if (<= t_3 2e+47)
       (fma 2.0 x1 (fma -6.0 x2 (* x1 -3.0)))
       (* (* (* x1 x1) (* x1 x1)) (- 6.0 (/ 3.0 x1)))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_3 <= -1.5e+167) {
		tmp = 8.0 * ((x1 * (x2 * x2)) / (1.0 + (x1 * x1)));
	} else if (t_3 <= 2e+47) {
		tmp = fma(2.0, x1, fma(-6.0, x2, (x1 * -3.0)));
	} else {
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (3.0 / x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	tmp = 0.0
	if (t_3 <= -1.5e+167)
		tmp = Float64(8.0 * Float64(Float64(x1 * Float64(x2 * x2)) / Float64(1.0 + Float64(x1 * x1))));
	elseif (t_3 <= 2e+47)
		tmp = fma(2.0, x1, fma(-6.0, x2, Float64(x1 * -3.0)));
	else
		tmp = Float64(Float64(Float64(x1 * x1) * Float64(x1 * x1)) * Float64(6.0 - Float64(3.0 / x1)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.5e+167], N[(8.0 * N[(N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+47], N[(2.0 * x1 + N[(-6.0 * x2 + N[(x1 * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1.50000000000000006e167

   1. Initial program 99.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around inf

      \[\leadsto \color{blue}{8 \cdot \frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 8 \cdot \color{blue}{\frac{x1 \cdot {x2}^{2}}{1 + {x1}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot {x2}^{2}}{\color{blue}{1 + {x1}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot {x2}^{2}}{\color{blue}{1} + {x1}^{2}} \]

      4. unpow2 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + {x1}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + {x1}^{2}} \]

      6. lower-+.f64 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + \color{blue}{{x1}^{2}}} \]

      7. pow2 N/A

         \[\leadsto 8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + x1 \cdot \color{blue}{x1}} \]

      8. lift-*.f64 66.1

         \[\leadsto 8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + x1 \cdot \color{blue}{x1}} \]

   4. Applied rewrites 66.1%

      \[\leadsto \color{blue}{8 \cdot \frac{x1 \cdot \left(x2 \cdot x2\right)}{1 + x1 \cdot x1}} \]

   if -1.50000000000000006e167 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 2.0000000000000001e47

   1. Initial program 99.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 93.5%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 91.1%

       \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right) \]

   if 2.0000000000000001e47 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 48.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x1 around inf

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{\color{blue}{x1}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 76.4

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{x1}\right) \]

   7. Applied rewrites 76.4%

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{\color{blue}{x1}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 80.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -1.5 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{x1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_2) (- t_2 3.0))
                (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
               t_1)
              (* t_0 t_2))
             (* (* x1 x1) x1))
            x1)
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))))
   (if (<= t_3 -1.5e+167)
     (fma 2.0 x1 (* 8.0 (* x1 (* x2 x2))))
     (if (<= t_3 2e+47)
       (fma 2.0 x1 (fma -6.0 x2 (* x1 -3.0)))
       (* (* (* x1 x1) (* x1 x1)) (- 6.0 (/ 3.0 x1)))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_3 <= -1.5e+167) {
		tmp = fma(2.0, x1, (8.0 * (x1 * (x2 * x2))));
	} else if (t_3 <= 2e+47) {
		tmp = fma(2.0, x1, fma(-6.0, x2, (x1 * -3.0)));
	} else {
		tmp = ((x1 * x1) * (x1 * x1)) * (6.0 - (3.0 / x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	tmp = 0.0
	if (t_3 <= -1.5e+167)
		tmp = fma(2.0, x1, Float64(8.0 * Float64(x1 * Float64(x2 * x2))));
	elseif (t_3 <= 2e+47)
		tmp = fma(2.0, x1, fma(-6.0, x2, Float64(x1 * -3.0)));
	else
		tmp = Float64(Float64(Float64(x1 * x1) * Float64(x1 * x1)) * Float64(6.0 - Float64(3.0 / x1)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.5e+167], N[(2.0 * x1 + N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+47], N[(2.0 * x1 + N[(-6.0 * x2 + N[(x1 * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision] * N[(6.0 - N[(3.0 / x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1.50000000000000006e167

   1. Initial program 99.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 63.8%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around inf

      \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot {x2}^{2}\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot {x2}^{2}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot {x2}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\right) \]

      4. lower-*.f64 63.8

         \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\right) \]

   10. Applied rewrites 63.8%

       \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\right) \]

   if -1.50000000000000006e167 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 2.0000000000000001e47

   1. Initial program 99.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 93.5%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 91.1%

       \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right) \]

   if 2.0000000000000001e47 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 48.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x1 around inf

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{\color{blue}{x1}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 76.4

         \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{x1}\right) \]

   7. Applied rewrites 76.4%

      \[\leadsto \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - \frac{3}{\color{blue}{x1}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 80.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -1.5 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + 6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_2) (- t_2 3.0))
                (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
               t_1)
              (* t_0 t_2))
             (* (* x1 x1) x1))
            x1)
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))))
   (if (<= t_3 -1.5e+167)
     (fma 2.0 x1 (* 8.0 (* x1 (* x2 x2))))
     (if (<= t_3 2e+47)
       (fma 2.0 x1 (fma -6.0 x2 (* x1 -3.0)))
       (+ x1 (* 6.0 (* (* x1 x1) (* x1 x1))))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_3 <= -1.5e+167) {
		tmp = fma(2.0, x1, (8.0 * (x1 * (x2 * x2))));
	} else if (t_3 <= 2e+47) {
		tmp = fma(2.0, x1, fma(-6.0, x2, (x1 * -3.0)));
	} else {
		tmp = x1 + (6.0 * ((x1 * x1) * (x1 * x1)));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	tmp = 0.0
	if (t_3 <= -1.5e+167)
		tmp = fma(2.0, x1, Float64(8.0 * Float64(x1 * Float64(x2 * x2))));
	elseif (t_3 <= 2e+47)
		tmp = fma(2.0, x1, fma(-6.0, x2, Float64(x1 * -3.0)));
	else
		tmp = Float64(x1 + Float64(6.0 * Float64(Float64(x1 * x1) * Float64(x1 * x1))));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.5e+167], N[(2.0 * x1 + N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+47], N[(2.0 * x1 + N[(-6.0 * x2 + N[(x1 * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(6.0 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1.50000000000000006e167

   1. Initial program 99.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 63.8%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around inf

      \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot {x2}^{2}\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot {x2}^{2}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot {x2}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\right) \]

      4. lower-*.f64 63.8

         \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\right) \]

   10. Applied rewrites 63.8%

       \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\right) \]

   if -1.50000000000000006e167 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 2.0000000000000001e47

   1. Initial program 99.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 93.5%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 91.1%

       \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right) \]

   if 2.0000000000000001e47 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 48.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around inf

      \[\leadsto x1 + \color{blue}{6 \cdot {x1}^{4}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x1 + 6 \cdot \color{blue}{{x1}^{4}} \]

      2. sqr-pow N/A

         \[\leadsto x1 + 6 \cdot \left({x1}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{x1}^{\left(\frac{4}{2}\right)}}\right) \]

      3. metadata-eval N/A

         \[\leadsto x1 + 6 \cdot \left({x1}^{2} \cdot {x1}^{\left(\frac{4}{2}\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto x1 + 6 \cdot \left({x1}^{2} \cdot {x1}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto x1 + 6 \cdot \left({x1}^{2} \cdot \color{blue}{{x1}^{2}}\right) \]

      6. pow2 N/A

         \[\leadsto x1 + 6 \cdot \left(\left(x1 \cdot x1\right) \cdot {\color{blue}{x1}}^{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto x1 + 6 \cdot \left(\left(x1 \cdot x1\right) \cdot {\color{blue}{x1}}^{2}\right) \]

      8. pow2 N/A

         \[\leadsto x1 + 6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot \color{blue}{x1}\right)\right) \]

      9. lift-*.f64 76.5

         \[\leadsto x1 + 6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot \color{blue}{x1}\right)\right) \]

   4. Applied rewrites 76.5%

      \[\leadsto x1 + \color{blue}{6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 80.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -1.5 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_2) (- t_2 3.0))
                (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
               t_1)
              (* t_0 t_2))
             (* (* x1 x1) x1))
            x1)
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))))
   (if (<= t_3 -1.5e+167)
     (fma 2.0 x1 (* 8.0 (* x1 (* x2 x2))))
     (if (<= t_3 2e+47)
       (fma 2.0 x1 (fma -6.0 x2 (* x1 -3.0)))
       (* 6.0 (* (* x1 x1) (* x1 x1)))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_3 <= -1.5e+167) {
		tmp = fma(2.0, x1, (8.0 * (x1 * (x2 * x2))));
	} else if (t_3 <= 2e+47) {
		tmp = fma(2.0, x1, fma(-6.0, x2, (x1 * -3.0)));
	} else {
		tmp = 6.0 * ((x1 * x1) * (x1 * x1));
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	tmp = 0.0
	if (t_3 <= -1.5e+167)
		tmp = fma(2.0, x1, Float64(8.0 * Float64(x1 * Float64(x2 * x2))));
	elseif (t_3 <= 2e+47)
		tmp = fma(2.0, x1, fma(-6.0, x2, Float64(x1 * -3.0)));
	else
		tmp = Float64(6.0 * Float64(Float64(x1 * x1) * Float64(x1 * x1)));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.5e+167], N[(2.0 * x1 + N[(8.0 * N[(x1 * N[(x2 * x2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+47], N[(2.0 * x1 + N[(-6.0 * x2 + N[(x1 * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -1.50000000000000006e167

   1. Initial program 99.8%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 63.8%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around inf

      \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot {x2}^{2}\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot {x2}^{2}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot {x2}^{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\right) \]

      4. lower-*.f64 63.8

         \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\right) \]

   10. Applied rewrites 63.8%

       \[\leadsto \mathsf{fma}\left(2, x1, 8 \cdot \left(x1 \cdot \left(x2 \cdot x2\right)\right)\right) \]

   if -1.50000000000000006e167 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 2.0000000000000001e47

   1. Initial program 99.2%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 93.5%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 91.1%

       \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right) \]

   if 2.0000000000000001e47 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 48.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around inf

      \[\leadsto \color{blue}{6 \cdot {x1}^{4}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 6 \cdot \color{blue}{{x1}^{4}} \]

      2. sqr-pow N/A

         \[\leadsto 6 \cdot \left({x1}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{x1}^{\left(\frac{4}{2}\right)}}\right) \]

      3. metadata-eval N/A

         \[\leadsto 6 \cdot \left({x1}^{2} \cdot {x1}^{\left(\frac{4}{2}\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto 6 \cdot \left({x1}^{2} \cdot {x1}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 6 \cdot \left({x1}^{2} \cdot \color{blue}{{x1}^{2}}\right) \]

      6. pow2 N/A

         \[\leadsto 6 \cdot \left(\left(x1 \cdot x1\right) \cdot {\color{blue}{x1}}^{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 6 \cdot \left(\left(x1 \cdot x1\right) \cdot {\color{blue}{x1}}^{2}\right) \]

      8. pow2 N/A

         \[\leadsto 6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot \color{blue}{x1}\right)\right) \]

      9. lift-*.f64 76.4

         \[\leadsto 6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot \color{blue}{x1}\right)\right) \]

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 80.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)\\ \mathbf{if}\;x1 \leq -1.85:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 6.0 (* (* x1 x1) (* x1 x1)))))
   (if (<= x1 -1.85)
     t_0
     (if (<= x1 1.4) (fma 2.0 x1 (fma -6.0 x2 (* x1 -3.0))) t_0))))

C

double code(double x1, double x2) {
	double t_0 = 6.0 * ((x1 * x1) * (x1 * x1));
	double tmp;
	if (x1 <= -1.85) {
		tmp = t_0;
	} else if (x1 <= 1.4) {
		tmp = fma(2.0, x1, fma(-6.0, x2, (x1 * -3.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(6.0 * Float64(Float64(x1 * x1) * Float64(x1 * x1)))
	tmp = 0.0
	if (x1 <= -1.85)
		tmp = t_0;
	elseif (x1 <= 1.4)
		tmp = fma(2.0, x1, fma(-6.0, x2, Float64(x1 * -3.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(6.0 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -1.85], t$95$0, If[LessEqual[x1, 1.4], N[(2.0 * x1 + N[(-6.0 * x2 + N[(x1 * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x1 < -1.8500000000000001 or 1.3999999999999999 < x1

   1. Initial program 40.4%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around inf

      \[\leadsto \color{blue}{6 \cdot {x1}^{4}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 6 \cdot \color{blue}{{x1}^{4}} \]

      2. sqr-pow N/A

         \[\leadsto 6 \cdot \left({x1}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{x1}^{\left(\frac{4}{2}\right)}}\right) \]

      3. metadata-eval N/A

         \[\leadsto 6 \cdot \left({x1}^{2} \cdot {x1}^{\left(\frac{4}{2}\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto 6 \cdot \left({x1}^{2} \cdot {x1}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 6 \cdot \left({x1}^{2} \cdot \color{blue}{{x1}^{2}}\right) \]

      6. pow2 N/A

         \[\leadsto 6 \cdot \left(\left(x1 \cdot x1\right) \cdot {\color{blue}{x1}}^{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 6 \cdot \left(\left(x1 \cdot x1\right) \cdot {\color{blue}{x1}}^{2}\right) \]

      8. pow2 N/A

         \[\leadsto 6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot \color{blue}{x1}\right)\right) \]

      9. lift-*.f64 88.0

         \[\leadsto 6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot \color{blue}{x1}\right)\right) \]

   4. Applied rewrites 88.0%

      \[\leadsto \color{blue}{6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)} \]

   if -1.8500000000000001 < x1 < 1.3999999999999999

   1. Initial program 99.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 86.4%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 73.2%

       \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot -3\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 72.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)\\ t_1 := \mathsf{fma}\left(2, x1, -3 \cdot x1\right)\\ \mathbf{if}\;x1 \leq -0.032:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x1 \leq -9 \cdot 10^{-79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x1 \leq 3.4 \cdot 10^{-109}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x1 \leq 1.4:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* 6.0 (* (* x1 x1) (* x1 x1)))) (t_1 (fma 2.0 x1 (* -3.0 x1))))
   (if (<= x1 -0.032)
     t_0
     (if (<= x1 -9e-79)
       t_1
       (if (<= x1 3.4e-109) (* -6.0 x2) (if (<= x1 1.4) t_1 t_0))))))

C

double code(double x1, double x2) {
	double t_0 = 6.0 * ((x1 * x1) * (x1 * x1));
	double t_1 = fma(2.0, x1, (-3.0 * x1));
	double tmp;
	if (x1 <= -0.032) {
		tmp = t_0;
	} else if (x1 <= -9e-79) {
		tmp = t_1;
	} else if (x1 <= 3.4e-109) {
		tmp = -6.0 * x2;
	} else if (x1 <= 1.4) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(6.0 * Float64(Float64(x1 * x1) * Float64(x1 * x1)))
	t_1 = fma(2.0, x1, Float64(-3.0 * x1))
	tmp = 0.0
	if (x1 <= -0.032)
		tmp = t_0;
	elseif (x1 <= -9e-79)
		tmp = t_1;
	elseif (x1 <= 3.4e-109)
		tmp = Float64(-6.0 * x2);
	elseif (x1 <= 1.4)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(6.0 * N[(N[(x1 * x1), $MachinePrecision] * N[(x1 * x1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * x1 + N[(-3.0 * x1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x1, -0.032], t$95$0, If[LessEqual[x1, -9e-79], t$95$1, If[LessEqual[x1, 3.4e-109], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[x1, 1.4], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x1 < -0.032000000000000001 or 1.3999999999999999 < x1

   1. Initial program 40.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around inf

      \[\leadsto \color{blue}{6 \cdot {x1}^{4}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 6 \cdot \color{blue}{{x1}^{4}} \]

      2. sqr-pow N/A

         \[\leadsto 6 \cdot \left({x1}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{x1}^{\left(\frac{4}{2}\right)}}\right) \]

      3. metadata-eval N/A

         \[\leadsto 6 \cdot \left({x1}^{2} \cdot {x1}^{\left(\frac{4}{2}\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto 6 \cdot \left({x1}^{2} \cdot {x1}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 6 \cdot \left({x1}^{2} \cdot \color{blue}{{x1}^{2}}\right) \]

      6. pow2 N/A

         \[\leadsto 6 \cdot \left(\left(x1 \cdot x1\right) \cdot {\color{blue}{x1}}^{2}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto 6 \cdot \left(\left(x1 \cdot x1\right) \cdot {\color{blue}{x1}}^{2}\right) \]

      8. pow2 N/A

         \[\leadsto 6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot \color{blue}{x1}\right)\right) \]

      9. lift-*.f64 87.7

         \[\leadsto 6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot \color{blue}{x1}\right)\right) \]

   4. Applied rewrites 87.7%

      \[\leadsto \color{blue}{6 \cdot \left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right)} \]

   if -0.032000000000000001 < x1 < -9.0000000000000006e-79 or 3.40000000000000012e-109 < x1 < 1.3999999999999999

   1. Initial program 99.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 91.8%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -3 \cdot x1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 39.8

         \[\leadsto \mathsf{fma}\left(2, x1, -3 \cdot x1\right) \]

   10. Applied rewrites 39.8%

       \[\leadsto \mathsf{fma}\left(2, x1, -3 \cdot x1\right) \]

   if -9.0000000000000006e-79 < x1 < 3.40000000000000012e-109

   1. Initial program 99.5%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2} \]
   3. Step-by-step derivation

      1. lower-*.f64 64.8

         \[\leadsto -6 \cdot \color{blue}{x2} \]

   4. Applied rewrites 64.8%

      \[\leadsto \color{blue}{-6 \cdot x2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 41.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(3 \cdot x1\right) \cdot x1\\ t_1 := x1 \cdot x1 + 1\\ t_2 := \frac{\left(t\_0 + 2 \cdot x2\right) - x1}{t\_1}\\ t_3 := x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot t\_2\right) \cdot \left(t\_2 - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot t\_2 - 6\right)\right) \cdot t\_1 + t\_0 \cdot t\_2\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(t\_0 - 2 \cdot x2\right) - x1}{t\_1}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-17}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;t\_3 \leq 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(2, x1, -3 \cdot x1\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+298}:\\ \;\;\;\;x1 + -6 \cdot x2\\ \mathbf{else}:\\ \;\;\;\;8 \cdot \left(\left(x1 \cdot x1\right) \cdot x2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (let* ((t_0 (* (* 3.0 x1) x1))
        (t_1 (+ (* x1 x1) 1.0))
        (t_2 (/ (- (+ t_0 (* 2.0 x2)) x1) t_1))
        (t_3
         (+
          x1
          (+
           (+
            (+
             (+
              (*
               (+
                (* (* (* 2.0 x1) t_2) (- t_2 3.0))
                (* (* x1 x1) (- (* 4.0 t_2) 6.0)))
               t_1)
              (* t_0 t_2))
             (* (* x1 x1) x1))
            x1)
           (* 3.0 (/ (- (- t_0 (* 2.0 x2)) x1) t_1))))))
   (if (<= t_3 -2e-17)
     (* -6.0 x2)
     (if (<= t_3 1e-15)
       (fma 2.0 x1 (* -3.0 x1))
       (if (<= t_3 2e+298) (+ x1 (* -6.0 x2)) (* 8.0 (* (* x1 x1) x2)))))))

C

double code(double x1, double x2) {
	double t_0 = (3.0 * x1) * x1;
	double t_1 = (x1 * x1) + 1.0;
	double t_2 = ((t_0 + (2.0 * x2)) - x1) / t_1;
	double t_3 = x1 + (((((((((2.0 * x1) * t_2) * (t_2 - 3.0)) + ((x1 * x1) * ((4.0 * t_2) - 6.0))) * t_1) + (t_0 * t_2)) + ((x1 * x1) * x1)) + x1) + (3.0 * (((t_0 - (2.0 * x2)) - x1) / t_1)));
	double tmp;
	if (t_3 <= -2e-17) {
		tmp = -6.0 * x2;
	} else if (t_3 <= 1e-15) {
		tmp = fma(2.0, x1, (-3.0 * x1));
	} else if (t_3 <= 2e+298) {
		tmp = x1 + (-6.0 * x2);
	} else {
		tmp = 8.0 * ((x1 * x1) * x2);
	}
	return tmp;
}

Julia

function code(x1, x2)
	t_0 = Float64(Float64(3.0 * x1) * x1)
	t_1 = Float64(Float64(x1 * x1) + 1.0)
	t_2 = Float64(Float64(Float64(t_0 + Float64(2.0 * x2)) - x1) / t_1)
	t_3 = Float64(x1 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(2.0 * x1) * t_2) * Float64(t_2 - 3.0)) + Float64(Float64(x1 * x1) * Float64(Float64(4.0 * t_2) - 6.0))) * t_1) + Float64(t_0 * t_2)) + Float64(Float64(x1 * x1) * x1)) + x1) + Float64(3.0 * Float64(Float64(Float64(t_0 - Float64(2.0 * x2)) - x1) / t_1))))
	tmp = 0.0
	if (t_3 <= -2e-17)
		tmp = Float64(-6.0 * x2);
	elseif (t_3 <= 1e-15)
		tmp = fma(2.0, x1, Float64(-3.0 * x1));
	elseif (t_3 <= 2e+298)
		tmp = Float64(x1 + Float64(-6.0 * x2));
	else
		tmp = Float64(8.0 * Float64(Float64(x1 * x1) * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := Block[{t$95$0 = N[(N[(3.0 * x1), $MachinePrecision] * x1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x1 * x1), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 + N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x1 + N[(N[(N[(N[(N[(N[(N[(N[(N[(2.0 * x1), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$2 - 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * N[(N[(4.0 * t$95$2), $MachinePrecision] - 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(x1 * x1), $MachinePrecision] * x1), $MachinePrecision]), $MachinePrecision] + x1), $MachinePrecision] + N[(3.0 * N[(N[(N[(t$95$0 - N[(2.0 * x2), $MachinePrecision]), $MachinePrecision] - x1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-17], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[t$95$3, 1e-15], N[(2.0 * x1 + N[(-3.0 * x1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+298], N[(x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision], N[(8.0 * N[(N[(x1 * x1), $MachinePrecision] * x2), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < -2.00000000000000014e-17

   1. Initial program 99.7%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2} \]
   3. Step-by-step derivation

      1. lower-*.f64 41.1

         \[\leadsto -6 \cdot \color{blue}{x2} \]

   4. Applied rewrites 41.1%

      \[\leadsto \color{blue}{-6 \cdot x2} \]

   if -2.00000000000000014e-17 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1.0000000000000001e-15

   1. Initial program 99.0%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 99.4%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -3 \cdot x1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 50.1

         \[\leadsto \mathsf{fma}\left(2, x1, -3 \cdot x1\right) \]

   10. Applied rewrites 50.1%

       \[\leadsto \mathsf{fma}\left(2, x1, -3 \cdot x1\right) \]

   if 1.0000000000000001e-15 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))))) < 1.9999999999999999e298

   1. Initial program 99.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
   3. Step-by-step derivation

      1. lower-*.f64 39.0

         \[\leadsto x1 + -6 \cdot \color{blue}{x2} \]

   4. Applied rewrites 39.0%

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]

   if 1.9999999999999999e298 < (+.f64 x1 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 (+.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) (-.f64 (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) #s(literal 3 binary64))) (*.f64 (*.f64 x1 x1) (-.f64 (*.f64 #s(literal 4 binary64) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64)))) #s(literal 6 binary64)))) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))) (*.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (/.f64 (-.f64 (+.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))) (*.f64 (*.f64 x1 x1) x1)) x1) (*.f64 #s(literal 3 binary64) (/.f64 (-.f64 (-.f64 (*.f64 (*.f64 #s(literal 3 binary64) x1) x1) (*.f64 #s(literal 2 binary64) x2)) x1) (+.f64 (*.f64 x1 x1) #s(literal 1 binary64))))))

   1. Initial program 30.3%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around -inf

      \[\leadsto \color{blue}{{x1}^{4} \cdot \left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x1}^{4} \cdot \color{blue}{\left(6 + -1 \cdot \frac{3 + -1 \cdot \frac{9 + 4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   4. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\left(\left(x1 \cdot x1\right) \cdot \left(x1 \cdot x1\right)\right) \cdot \left(6 - 1 \cdot \frac{3 - 1 \cdot \frac{9 - -4 \cdot \left(2 \cdot x2 - 3\right)}{x1}}{x1}\right)} \]

   5. Taylor expanded in x2 around inf

      \[\leadsto 8 \cdot \color{blue}{\left({x1}^{2} \cdot x2\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 8 \cdot \left({x1}^{2} \cdot \color{blue}{x2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto 8 \cdot \left({x1}^{2} \cdot x2\right) \]

      3. pow2 N/A

         \[\leadsto 8 \cdot \left(\left(x1 \cdot x1\right) \cdot x2\right) \]

      4. lift-*.f64 38.4

         \[\leadsto 8 \cdot \left(\left(x1 \cdot x1\right) \cdot x2\right) \]

   7. Applied rewrites 38.4%

      \[\leadsto 8 \cdot \color{blue}{\left(\left(x1 \cdot x1\right) \cdot x2\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 23: 32.3% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x2 \leq -1.85 \cdot 10^{-121}:\\ \;\;\;\;-6 \cdot x2\\ \mathbf{elif}\;x2 \leq 1.06 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{fma}\left(2, x1, -3 \cdot x1\right)\\ \mathbf{else}:\\ \;\;\;\;x1 + -6 \cdot x2\\ \end{array} \end{array} \]

FPCore

(FPCore (x1 x2)
 :precision binary64
 (if (<= x2 -1.85e-121)
   (* -6.0 x2)
   (if (<= x2 1.06e-156) (fma 2.0 x1 (* -3.0 x1)) (+ x1 (* -6.0 x2)))))

C

double code(double x1, double x2) {
	double tmp;
	if (x2 <= -1.85e-121) {
		tmp = -6.0 * x2;
	} else if (x2 <= 1.06e-156) {
		tmp = fma(2.0, x1, (-3.0 * x1));
	} else {
		tmp = x1 + (-6.0 * x2);
	}
	return tmp;
}

Julia

function code(x1, x2)
	tmp = 0.0
	if (x2 <= -1.85e-121)
		tmp = Float64(-6.0 * x2);
	elseif (x2 <= 1.06e-156)
		tmp = fma(2.0, x1, Float64(-3.0 * x1));
	else
		tmp = Float64(x1 + Float64(-6.0 * x2));
	end
	return tmp
end

Wolfram

code[x1_, x2_] := If[LessEqual[x2, -1.85e-121], N[(-6.0 * x2), $MachinePrecision], If[LessEqual[x2, 1.06e-156], N[(2.0 * x1 + N[(-3.0 * x1), $MachinePrecision]), $MachinePrecision], N[(x1 + N[(-6.0 * x2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x2 < -1.8500000000000001e-121

   1. Initial program 70.1%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto \color{blue}{-6 \cdot x2} \]
   3. Step-by-step derivation

      1. lower-*.f64 31.7

         \[\leadsto -6 \cdot \color{blue}{x2} \]

   4. Applied rewrites 31.7%

      \[\leadsto \color{blue}{-6 \cdot x2} \]

   if -1.8500000000000001e-121 < x2 < 1.06e-156

   1. Initial program 69.5%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x2 around 0

      \[\leadsto \color{blue}{2 \cdot x1 + \left(3 \cdot \frac{{x1}^{2} \cdot \left(3 \cdot {x1}^{2} - x1\right)}{1 + {x1}^{2}} + \left(3 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} + \left(x2 \cdot \left(\left(6 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} + \left(8 \cdot \frac{x1 \cdot x2}{1 + {x1}^{2}} + \left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(2 \cdot \left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) + 2 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}}\right)}{1 + {x1}^{2}} + 8 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}}\right)\right)\right) - 6 \cdot \frac{1}{1 + {x1}^{2}}\right) + \left(\left(1 + {x1}^{2}\right) \cdot \left(2 \cdot \frac{x1 \cdot \left(\left(3 \cdot \frac{{x1}^{2}}{1 + {x1}^{2}} - \left(3 + \frac{x1}{1 + {x1}^{2}}\right)\right) \cdot \left(3 \cdot {x1}^{2} - x1\right)\right)}{1 + {x1}^{2}} + {x1}^{2} \cdot \left(4 \cdot \frac{3 \cdot {x1}^{2} - x1}{1 + {x1}^{2}} - 6\right)\right) + {x1}^{3}\right)\right)\right)\right)} \]

   4. Applied rewrites 69.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x1, \mathsf{fma}\left(3, \frac{\left(x1 \cdot x1\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)}{1 + x1 \cdot x1}, \mathsf{fma}\left(3, \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(x2, \mathsf{fma}\left(6, \frac{x1 \cdot x1}{1 + x1 \cdot x1}, \mathsf{fma}\left(8, \frac{x1 \cdot x2}{1 + x1 \cdot x1}, \left(1 + x1 \cdot x1\right) \cdot \mathsf{fma}\left(2, \frac{x1 \cdot \mathsf{fma}\left(2, 3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right), 2 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1}\right)}{1 + x1 \cdot x1}, 8 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1}\right)\right)\right) - 6 \cdot \frac{1}{1 + x1 \cdot x1}, \mathsf{fma}\left(1 + x1 \cdot x1, \mathsf{fma}\left(2, \frac{x1 \cdot \left(\left(3 \cdot \frac{x1 \cdot x1}{1 + x1 \cdot x1} - \left(3 + \frac{x1}{1 + x1 \cdot x1}\right)\right) \cdot \left(3 \cdot \left(x1 \cdot x1\right) - x1\right)\right)}{1 + x1 \cdot x1}, \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{3 \cdot \left(x1 \cdot x1\right) - x1}{1 + x1 \cdot x1} - 6\right)\right), \left(x1 \cdot x1\right) \cdot x1\right)\right)\right)\right)\right)} \]

   5. Taylor expanded in x1 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -6 \cdot x2 + x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 49.5%

      \[\leadsto \mathsf{fma}\left(2, x1, \mathsf{fma}\left(-6, x2, x1 \cdot \left(x2 \cdot \left(8 \cdot x2 - 12\right) - 3\right)\right)\right) \]

   8. Taylor expanded in x2 around 0

      \[\leadsto \mathsf{fma}\left(2, x1, -3 \cdot x1\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 36.2

         \[\leadsto \mathsf{fma}\left(2, x1, -3 \cdot x1\right) \]

   10. Applied rewrites 36.2%

       \[\leadsto \mathsf{fma}\left(2, x1, -3 \cdot x1\right) \]

   if 1.06e-156 < x2

   1. Initial program 69.5%

      \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

   2. Taylor expanded in x1 around 0

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
   3. Step-by-step derivation

      1. lower-*.f64 30.0

         \[\leadsto x1 + -6 \cdot \color{blue}{x2} \]

   4. Applied rewrites 30.0%

      \[\leadsto x1 + \color{blue}{-6 \cdot x2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 26.1% accurate, 46.3× speedup

Math

\[\begin{array}{l} \\ -6 \cdot x2 \end{array} \]

FPCore

(FPCore (x1 x2) :precision binary64 (* -6.0 x2))

C

double code(double x1, double x2) {
	return -6.0 * x2;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x1, x2)
use fmin_fmax_functions
    real(8), intent (in) :: x1
    real(8), intent (in) :: x2
    code = (-6.0d0) * x2
end function

Java

public static double code(double x1, double x2) {
	return -6.0 * x2;
}

Python

def code(x1, x2):
	return -6.0 * x2

Julia

function code(x1, x2)
	return Float64(-6.0 * x2)
end

MATLAB

function tmp = code(x1, x2)
	tmp = -6.0 * x2;
end

Wolfram

code[x1_, x2_] := N[(-6.0 * x2), $MachinePrecision]

Derivation

1. Initial program 69.7%

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \]

2. Taylor expanded in x1 around 0

   \[\leadsto \color{blue}{-6 \cdot x2} \]
3. Step-by-step derivation

   1. lower-*.f64 26.1

      \[\leadsto -6 \cdot \color{blue}{x2} \]

4. Applied rewrites 26.1%

   \[\leadsto \color{blue}{-6 \cdot x2} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025114 
(FPCore (x1 x2)
  :name "Rosa's FloatVsDoubleBenchmark"
  :precision binary64
  (+ x1 (+ (+ (+ (+ (* (+ (* (* (* 2.0 x1) (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))) (- (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)) 3.0)) (* (* x1 x1) (- (* 4.0 (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))) 6.0))) (+ (* x1 x1) 1.0)) (* (* (* 3.0 x1) x1) (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))) (* (* x1 x1) x1)) x1) (* 3.0 (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))))))